SOLUTION: A group of friends planned a trip and chartered a bus for 300$. When 3 more friends decided to join the trip, the cost for the bus was $5 less per person. In all, how many people m

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: A group of friends planned a trip and chartered a bus for 300$. When 3 more friends decided to join the trip, the cost for the bus was $5 less per person. In all, how many people m      Log On


   

Question 697541: A group of friends planned a trip and chartered a bus for 300$. When 3 more friends decided to join the trip, the cost for the bus was $5 less per person. In all, how many people made the trip.
Answer by Stitch(470) About Me  (Show Source):
You can put this solution on YOUR website!
Set Up:
Let A = the number of friends in the begining.
Let B = the cost per person
-----------------------------
Equation 1: B+=+300%2FA
Equation 2: B+-+5+=+300%2F%28A+%2B+3%29
Plug (300/A) into equation 2 for B
Equation 2: B+-+5+=+300%2F%28A+%2B+3%29
%28300%2FA%29+-+5+=+300%2F%28A%2B3%29
To simplify the equation, multiply both sides by (A+3)
%28300%2FA%29%2A%28A%2B3%29+-+5%2A%28A%2B3%29+=+300
Multiply both sides by A
300%2A%28A%2B3%29+-+5%2A%28A%2B3%29%2AA+=+300A
Simplify
300A+%2B+900+-+5A%2A%28A%2B3%29+=+300A
Simplify
300A+%2B+900+-+5A%5E2+-+15A+=+300A
Combine like terms
285A+%2B+900+-+5A%5E2+=+300A
Subtract 285A from both sides
900+-+5A%5E2+=+15A
Add 5A^2 to both sides
900+=+5A%5E2+%2B+15A
*This step is not required but to make the numbers smaller pull a 5 from the right side of the equation
900+=+5%2A%28A%5E2+%2B+3A%29
Divide both sides by 5
180+=+A%5E2+%2B+3A
Subtract 180 from both sides
0+=+A%5E2+%2B+3A+-+180
Use the quadratic equation.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aA%5E2%2BbA%2Bc=0 (in our case 1A%5E2%2B3A%2B-180+=+0) has the following solutons:

A%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%283%29%5E2-4%2A1%2A-180=729.

Discriminant d=729 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-3%2B-sqrt%28+729+%29%29%2F2%5Ca.

A%5B1%5D+=+%28-%283%29%2Bsqrt%28+729+%29%29%2F2%5C1+=+12
A%5B2%5D+=+%28-%283%29-sqrt%28+729+%29%29%2F2%5C1+=+-15

Quadratic expression 1A%5E2%2B3A%2B-180 can be factored:
1A%5E2%2B3A%2B-180+=+1%28A-12%29%2A%28A--15%29
Again, the answer is: 12, -15. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-180+%29

Note that there are two answers; 12 & -15.
Now we can not have negative people so your only answer is 12.
That means that 12 people were the original group and then 3 more people joined.
So there were 15 total people that made the trip.